A step-by-step Chebyshev space-time spectral method for force vibration of functionally graded structures

This paper proposes a new step-by-step Chebyshev space-time spectral method to analyze the force vibration of functionally graded material structures. Although traditional space-time spectral methods can reduce the accuracy mismatch between temporal low-order finite difference and spatial high-order discretization, their time collocation points must increase dramatically to solve highly oscillatory solutions of structural vibration, which results in a surge in computing time and a decrease in accuracy. To address this problem, we introduced the step-by-step idea in the space-time spectral method. The Chebyshev polynomials and Lagrange’s equation were applied to derive discrete spatial governing equations, and a matrix projection method was used to map the calculation results of previous steps as the initial conditions of the subsequent steps. A series of numerical experiments were carried out. The results of the proposed method were compared with those obtained by traditional space-time spectral methods, which showed that higher accuracy could be achieved in a shorter computation time than the latter in highly oscillatory cases.